Finite Group Databasebeta

← #20282718

Subgroups — #20282718

Order 6784.

578 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133199C211yesmaximal, characteristic, complemented
2C2×#13133192C211yesmaximal, characteristic, complemented
3#16810561C211yesmaximal, characteristic, complemented
4#16810545C212yesmaximal, complemented
5#16810541C211yesmaximal, characteristic, complemented
6#16810540C212yesmaximal, complemented
7#16810538C212yesmaximal, complemented
8#16810532C211yesmaximal, characteristic, complemented
9#16810532C212yesmaximal, complemented
10#16810482C211yesmaximal, characteristic, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C22×D212C2212yes—
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15C22×C22⋊D53C2212yes—
16C22×C4×D53C2211yescharacteristic
17#13133199—24no—
18#13133192—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×C22⋊Dic53C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29#13133124C2212yes—
30#13133124C2211yescharacteristic
31#13133124C2212yes—
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2312yes—
39C23×D53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—22no—
45C2×C22⋊D53—42no—
46C2×C22⋊D53—42no—
47C2×C22⋊D53—42no—
48C2×C22⋊D53—42no—
49C22×Dic53C2312yes—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C2×D212—24no—
54C2×D212—22no—
55C2×D212—42no—
56C2×D212—42no—
57C2×D212—42no—
58C2×C4×D53—42no—
59C2×C4×D53D414yescomplemented
60C4⋊C4×C53—22no—
61C4⋊C4×C53—42no—
62C22⋊C4×C53—22no—
63C22⋊C4×C53—42no—
64C22⋊Dic53—22no—
65#11719921—22no—
66#11719921—24no—
67#11719921—42no—
68#11719921—42no—
69#11719921—42no—
70#11719921—42no—
71C4⋊Dic53—22no—
72#11719919—22no—
73#11719919—42no—
74C23×C53—21no—
75C23×C53—22no—
76C23×C53—21no—
77C23×C53—22no—
78C23×C53—21no—
79C23×C53C2411yescharacteristic
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—21no—
83C22×D53—21no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53C2×D412yes—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53C2×D412yes—
102C22⋊D53—44no—
103C22⋊D53—48no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53C2×D412yes—
114D212—44no—
115D212—48no—
116C4×D53—44no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53C22×D411yescharacteristic
123C22×C53H(5,𝔽2)12yes—
124C22×C53H(5,𝔽2)12yes—
125C22×C53H(5,𝔽2)11yescharacteristic
126C22×C532−1+411yescharacteristic
127C2×D53—22no—
128C2×D53—22no—
129C2×D53—22no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C4×C53—22no—
140C4×C53—42no—
141C4×C53—41no—
142C4×C53—42no—
143Dic53—22no—
144Dic53—41no—
145Dic53—41no—
146Dic53—42no—
147#2813—531nomaximal, Sylow
148C2×C53—81no—
149C2×C53#53411yescharacteristic
150C2×C53#53511yescharacteristic
151C2×C53#53711yescharacteristic
152C2×C53#56012yes—
153C2×C53#56212yes—
154D53—42no—
155D53—82no—
156C2×#123—531no—
157C2×#121—531no—
158#400—531no—
159#399D5311yescharacteristic, complemented
160#396—532no—
161#394—532no—
162#394—532no—
163#394—531no—
164#392—531no—
165#392—532no—
166#380—531no—
167C53#281311yescharacteristic, complemented, Sylow
168C22×D4—531no—
169C22×D4—532no—
170C22×D4—532no—
171C22×D4—531no—
172C23×C4—531no—
173#123—1064no—
174#121—1064no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—532no—
178C2×C4⋊C4C2×D5311yescharacteristic
179C2×C4⋊C4C2×D5312yes—
180C2×C22⋊C4—531no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—531no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4C2×D5311yescharacteristic
188C2×C22⋊C4C2×D5312yes—
189#95—531no—
190#95—532no—
191#95—532no—
192#95C2×D5311yescharacteristic
193C24—531no—
194C24—532no—
195C24C22×D5311yescharacteristic
196C2×D4—1064no—
197C2×D4—1062no—
198C2×D4—1062no—
199C2×D4—2122no—
200C2×D4—2122no—
201C2×D4—2122no—
202C2×D4—2122no—
203C2×D4—2122no—
204C2×D4—2122no—
205C2×D4—2122no—
206C22×C4—531no—
207C22×C4—534no—
208C22×C4—532no—
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—2122no—
212C22×C4C22×D5312yes—
213C22×C4C22×D5312yes—
214C22×C4C22×D5311yescharacteristic
215C22×C4C22×D5311yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—1062no—
219C4⋊C4—1062no—
220C4⋊C4—2122no—
221C22⋊C4—22no—
222C22⋊C4—42no—
223C22⋊C4—1064no—
224C22⋊C4—1062no—
225C22⋊C4—1062no—
226C22⋊C4—2122no—
227C22⋊C4—2122no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C23—21no—
231C23—22no—
232C23—21no—
233C23—22no—
234C23—21no—
235C23—532no—
236C23—1062no—
237C23—1061no—
238C23—1062no—
239C23—1061no—
240C23—1062no—
241C23—1062no—
242C23—1062no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23—1062no—
247C23C23×D5311yescharacteristic
248D4—2124no—
249D4—2128no—
250D4—2124no—
251D4—2128no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—532no—
262C2×C4—1062no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1062no—
266C2×C4—1061no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1061no—
271C2×C4—2124no—
272C2×C4D4×D5312yes—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—1062no—
279C22—1062no—
280C22—1062no—
281C22—2122no—
282C22—2122no—
283C22—2122no—
284C22—2122no—
285C22—2122no—
286C22—2122no—
287C22—2122no—
288C22—2122no—
289C22—2122no—
290C22C2×D4×D5311yescharacteristic
291C22#1313326212yes—
292C22#1313326311yescharacteristic
293C22#1313326312yes—
294C22#1313326511yescharacteristic
295C4—22no—
296C4—42no—
297C4—42no—
298C4—41no—
299C4—1062no—
300C4—2122no—
301C4—2121no—
302C4—2121no—
303C2—81no—
304C2—4242no—
305C2—2122no—
306C2#1681118411yescharacteristic
307C2#1681118611yescharacteristic
308C2#1681120311yescharacteristic
309C2#1681136312yes—
310C2#1681136912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2813
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#13133199C2semidirect
2. C2×#13133192C2semidirect
3. #16810561C2semidirect
4. #16810545C2semidirect
5. #16810541C2semidirect
6. #16810540C2semidirect
7. #16810538C2semidirect
8. #16810532C2semidirect
9. #16810532C2semidirect
10. #16810482C2semidirect
11. C53×#399C2semidirect
59. C2×C4×D53D4semidirect
159. #399D53semidirect
167. C53#2813semidirect